Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

Negative Numbers

Stage: 3

Article by Jill Howard

Published May 2009,January 2008,February 2011.

Nowadays we use negative numbers in many contexts and, as a result,
they seem perfectly natural to us. That's because we've been taught
to see numbers as a continuous number line, stretching out from
zero in both the positive and negative directions. To us, -3 is
just as real as +3 is, but this was not always the case. Negative
numbers have only fairly recently become accepted as part of the
system of numbers that mathematicians are allowed to use. While a
great deal of very advanced maths was developed by ancient
civilisations, mathematicians in most cultures had no understanding
of what a negative number could mean. In this article we're going
to explore some of the earliest appearances of negative numbers and
how attitudes towards them have changed over the centuries.

Among the earliest people to use negative numbers in calculations
were the ancient Chinese. They used counting rods to perform
calculations, with red rods for positive numbers and black rods for
negative numbers. The example below shows some Chinese numerals
represented by rods, and the diagram on the right shows which
numbers these symbols represent.

Indian mathematicians also used negative numbers long before
Western civilisations. An ancient manuscript from 200 BC shows that
they used to use the + sign that we now associate with addition and
positive numbers, to denote a negative number. Although negative
numbers were used in calculations, negative answers to mathematical
problems were usually considered meaningless and were
discarded.

The ancient Greeks also dismissed any solutions to equations
that came out negative. They called them "absurd" and "impossible"
and completely ignored them. They couldn't see how a negative
answer could be meaningful, because it was not possible to have a
quantity that was less than
nothing . This opinion was passed down to later
mathematicians in Europe for more than a thousand years, so very
little progress in negative number arithmetic was made for a long
time.

Can you think of something in the real world where you can
have a negative quantity that actually means something? Today we
are quite familiar with the idea of somebody being in debt and
therefore having a negative amount of money. It means that they
have no money in their possession, and actually owe someone else
money as well. It's not a very nice position to be in, but debt is
a form of negative quantity that has been around for thousands of
years. In fact calculations involving money were the only ones that
were allowed to have negative answers, but most mathematicians
weren't interested in that kind of problem.

So, even though everyone was quite happy to allow subtraction, and
could understand the notion of debt, it took centuries before
mathematicians understood or accepted that negative numbers could
exist as genuine numbers in their own right. In 1759 Francis
Meseres wrote that negative numbers:

"darken the very whole doctrines
of the equations and to make dark of the things which are in their
nature excessively obvious and simple. It would have been desirable
in consequence that the negative roots were never allowed in
algebra or that they were discarded" .

Even as late as 1803 the famous French mathematician Carnot was
worried about the reality of negative numbers:

"to really obtain an isolated
negative quantity, it would be necessary to cut off an effective
quantity from zero, to remove something of nothing: impossible
operation. How thus to conceive an isolated negative
quantity?"

Some mathematicians in the 17th century discovered that negative
numbers did have their uses. Provided they didn't worry about what
negative numbers meant, and more particularly what the square roots
of negative numbers meant, they found that they could solve some
very tricky equations, like cubic and quartic equations. What's
more, although the intermediate steps of a calculation may have
involved negative numbers, the solution often came out as a real,
positive number which was exactly what they wanted.

Since then mathematicians and scientists have found all sorts of
uses for negative numbers. We now recognise that in many cases a
negative answer can be a real, meaningful solution and can be
thought of in terms of direction . For instance, if I wanted
to calculate how many steps forward Robert has taken, and the
answer is -5, then it means he has taken 5 steps backwards. The
first person to recognise the link between negative numbers and
direction was John Wallis, a mathematician in the 17th century. He
was the first to come up with the idea of a number line as a
geometrical representation of the number system. Confusingly
however, he also thought that negative numbers were larger than
infinity!

Nowadays we use negative numbers just like any other numbers
without even a second thought. Their troubled history shows how the
simple mathematical principles we take for granted have taken
thousands of years to develop. Physical meaning has given way to
algebraic utility, but negative numbers and their derivatives have
turned out to have all kinds of practical applications. Take the
square root of -1 for example - it seems meaningless in itself, but
many calculations in science and engineering wouldn't be possible
without it.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.